324 Sir W. Rowan Hamilton o?i Quaieniiofis. 



R^R/(..)RH'R"R;"; 



r/r;'(..)R'R"R"'I^/^; 

 R/'R/"(..)R"ii"'R'^R/; }• ' • iQ'"') 

 r;''r/v(..)R"'R'^'Rr;; 



R/vR^(..)R'^RR'R/'. J 



These five formulae establish a remarkable connexion he- 

 tvoeen one sj^herical pentagon and another {when constructed 

 according to the foregoing rules), through the medium oi'Jive 

 spherical C07iics\ of which five curves each touches two sides 

 of one pentagon and has its foci at two corners of the other. 

 If we suppose for simplicity that each of the ten moduli is 

 = 1, the dependence of six quaternions by multiplication on 

 four (as their three binary, two ternary, and one quaternary 

 product, all taken without altering the order of succession of 

 the factors) will give eighteen distinct equations between the 

 ten amplitudes and the twenty polar coordinates of the ten 

 quaternions here considered ; it is therefore in general per- 

 mitted to assume at pleasure twelve of these coordinates, or 

 to choose six of the ten points upon the sphere. Not only, 

 therefore, may we in general take one of the two pentagons 

 arbitrarily^ but also, at the same time, may assume one cor- 

 ner of the other pentagon (subject of course to exceptional 

 cases); and, after a suitable choice of the ten amplitudes, the 

 five relations (Q'".), between the two pentagons and the five 

 conies, will still hold good. 



17. A very particular (or rather limiting) yet not inelegant 

 case of this theorem is furnished by the consideration of the 

 plane and regular pentagon of elementary geometry, as com- 

 pared with that other and interior pentagon which is deter- 

 mined by the intersections of its five diagonals. Denoting 

 by Ry that corntjr of the interior pentagon which is nearest to 

 the side R R' of the exterior one; by R/ that corner which is 

 nearest to R' R", and so on to R/^ ; the relations (Q'".) are 

 satisfied, the symbol ( . . ) now denoting that the two points 

 written before it are foci of an ordinary (or plane) ellipse, in- 

 scribed in the plane quadrilateral whose corners are the four 

 points written after it. We may add, that (in this particular 

 case) two points of contact for each of the five quadrilaterals 

 are corners of the interior pentagon; and that the axis major 

 of each of the five inscribed ellipses is equal to a side of the 

 exterior figure. 



[To be continued.] 



